Question: Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{z^2 - 25}{z - 5}$
Explanation: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = z$ $ b = \sqrt{25} = -5$ So we can rewrite the expression as: $a = \dfrac{({z} {-5})({z} + {5})} {z - 5} $ We can divide the numerator and denominator by $(z - 5)$ on condition that $z \neq 5$ Therefore $a = z + 5; z \neq 5$